A. It is hard to believe that a climate variable is the only
possible instrument for national identity. Why not take a look at
some of the World Governance Index (WGI) or other cross-country
variables published by the World Bank - these vary over time.

B. Even if this is the only possible instrument, you seem to be
forgetting that -xtivreg- and other programs such as -ivreg2-
automatically include all exogenous variables as instruments. The
steps that you describe - matching the climate variable with all
observations for each country and then using xtivreg - should
work. I have no idea whether the results will be satisfactory.